Let Ω ⊂  C be a non-empty open connected set and O < p < ∞. Define H^p(Ω) to be the set of all analytic functions f in Ω such that there exists a harmonic function u_f in Ω with |f|^p  < u_f· Let K ⊂ Ω be compact and such that Ω ∖, K is connected. Then the set K is said to be a removable singularity for H^p(Ω ∖ K) if H^p(Ω)= H^p(Ω  ∖ K). Hejhal proved in 1973 that this notion does not depend on Ω.

In this thesis we give a survey of the theory of removable singularities for Hardy spaces. We use potential theory, conformal mappings, harmonic measures and Banach space techniques to give new results. One new result is that if dim K > min {1, p} then K is not removable. Several theorems about removability of sets lying on rectifiable curves and also conditions for removability of some planar self-similar Cantor sets are given.